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V.: Gaussian multiplicative chaos and applications: a review, arxiv
"... In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of th ..."
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In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2dLiouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from finance, through the KolmogorovObukhov model of turbulence to 2dLiouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.
On the heat kernel and the Dirichlet form of Liouville Brownian Motion
"... In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville qua ..."
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In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially [13] and the techniques introduced in [14]. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a nontrivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in [14] was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular, the theory developed for example in [28, 29, 30], whose aim is to capture the “geometry ” of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.
Complex Gaussian multiplicative chaos
"... In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the ..."
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In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the socalled Tachyon fields, the conformally invariant operators in critical Liouville Quantum
3 On the heat kernel and the Dirichlet form of Liouville Brownian Motion
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unknown title
, 2014
"... We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non trivial lower and upper bounds. ..."
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We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non trivial lower and upper bounds.
Geodesic distances in quantum Liouville gravity
"... In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d ..."
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In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.